Finding distribution given bivariate normal $f_{xy}$

Solution 1:

That is right. There is a more general way to calculate such affine transformations you probably will see later. We have already established that $$ \begin{pmatrix} X\\ Y \end{pmatrix} \sim N_2\left( \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 1& \rho\\ \rho&1 \end{pmatrix}\right) $$ then $aX+bY+c = (a,b)(X,Y)^T + c$ and we find $$ aX+bY+c \sim N\left( c + (a,b) \begin{pmatrix} 0 \\ 0 \end{pmatrix}, (a,b) \begin{pmatrix} 1& \rho\\ \rho&1 \end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix} \right) = N(c, a^2+b^2 +2ab\rho). $$ This is from a general calculation of affine transformation $Y = \eta + BX$, where $X\sim N_p(\mu,\Sigma)$ is p-dimensional normal distributed, $B$ is a $k\times p$ matrix and $\eta\in \mathbb{R}^k$. Then $$Y \sim N_k(\eta+B\mu, B\, \Sigma \,B^T). $$